Half-life: distinguish the amount remaining from the amount lost
Use matching time units and interpret repeated halvings correctly.
Method
A constant half-life means the same fraction disappears during each equal time interval, not the same amount. After one half-life, half remains; after two, one quarter remains. Divide elapsed time by the half-life to obtain the exponent. Both times must use the same unit. Fractional exponents represent times between complete halvings.
Worked example
An initial quantity of 80 with a half-life of 6 hours has 20 remaining after 12 hours. Enter 80, 6 and 12 in Half-life decay. The lost amount is 60, not 20, and the remaining fraction is 25%. The example uses abstract quantities, so the result retains whatever quantity unit the input represents.
A0 = 80; h = 6 h; t = 12 h
A = 80 × (1/2)^(12/6) = 20
A0 − A = 60; A / A0 × 100 = 25%
t = h × ln(A / A0) / ln(1/2)
Checks and limits
The logarithmic equation below finds time from a positive target smaller than the initial quantity. A zero target has no finite time in this ideal model. A changing decay rate or new material entering the system requires a different model. These educational calculations do not estimate medication dosing, radiation exposure or a safe handling period.
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Source: OpenStax